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GradeEquation of Tangent of a Parabola

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If the chords of rectangular hyperbola \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\]touches the parabola \[{{y}^{2}}=4ax\]then the locus of their mid – points is

(a) \[{{x}^{2}}\left( y-a \right)={{y}^{3}}\]

(b) \[{{y}^{2}}\left( x-a \right)={{x}^{3}}\]

(c) \[x\left( y-a \right)=y\]

(d) \[y\left( x-a \right)=x\]

(a) \[{{x}^{2}}\left( y-a \right)={{y}^{3}}\]

(b) \[{{y}^{2}}\left( x-a \right)={{x}^{3}}\]

(c) \[x\left( y-a \right)=y\]

(d) \[y\left( x-a \right)=x\]

The circle drawn with variable chord \[x+ay-5=0\] (\[a\] being parameter) of the parabola \[{{y}^{2}}=20x\] as diameter will always touch the line

(a) \[x+5=0\]

(b) \[y+5=0\]

(c) \[x+y+5=0\]

(d) \[x-y+5=0\]

(a) \[x+5=0\]

(b) \[y+5=0\]

(c) \[x+y+5=0\]

(d) \[x-y+5=0\]

Find the equation of the tangent to the parabola \[{y^2} = 5x\], that is parallel to \[y = 4x + 1\] which meets the parabola at the coordinate \[\left( {\dfrac{5}{{64}},\dfrac{5}{8}} \right)\].

If the tangent to the parabola ${{y}^{2}}=4ax$ meets the axis in T and tangent at the vertex A in Y and the rectangle TAYG is completed, then the locus of G is

(a) ${{y}^{2}}+2ax=0$

(b) ${{y}^{2}}+ax=0$

(c) ${{x}^{2}}+ay=0$

(d) None of these

(a) ${{y}^{2}}+2ax=0$

(b) ${{y}^{2}}+ax=0$

(c) ${{x}^{2}}+ay=0$

(d) None of these

A curve has equation \[y={{x}^{2}}-4x+4\] and a line has equation\[y=mx\], where \[m\] is a constant.

1. For the case where \[m=1\], the curve and the line intersect at the points A and B. Find the coordinates of the midpoint of AB.

2. Find the non-zero value of \[m\] for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.

1. For the case where \[m=1\], the curve and the line intersect at the points A and B. Find the coordinates of the midpoint of AB.

2. Find the non-zero value of \[m\] for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.

How do you find the equations of both lines through point \[\left( {2, - 3} \right)\] that are tangent to the parabola \[y = {x^2} + x\]?

How do you find the equation of the tangent line to the graph of $f\left( x \right)={{x}^{2}}+1$ at point $\left( 2,5 \right)$.

How do you find the equation of the tangent line to the graph of $f\left( x \right)={{x}^{2}}+1$ at point $\left( 2,5 \right)$.

How do you find the equation of the tangent line to the graph of $f\left( x \right)={{x}^{2}}+1$ at point $\left( 2,5 \right)$.

The focal chord to ${y^2} = 64x$ is a tangent to ${\left( {x - 4} \right)^2} + {\left( {y - 2} \right)^2} = 4$ then the possible values of the slope of this chord is

a.$0, - \dfrac{{12}}{{35}}$

b.$0,\dfrac{{12}}{{35}}$

c.$0, - \dfrac{{35}}{{12}}$

d.$0, - \dfrac{6}{{35}}$

a.$0, - \dfrac{{12}}{{35}}$

b.$0,\dfrac{{12}}{{35}}$

c.$0, - \dfrac{{35}}{{12}}$

d.$0, - \dfrac{6}{{35}}$

Let PQ be the focal chord of the parabola $ {{y}^{2}}=4ax $ . The tangent to the parabola at P and Q meets at point lying on the line y = 2x + a, a < 0.

If chord PQ subtends an angle $ \theta $ at the vertex of $ {{y}^{2}}=4ax $ , then $ \tan \theta = $

a). $ \dfrac{2\sqrt{7}}{3} $

b). $ -\dfrac{2\sqrt{7}}{3} $

c). $ \dfrac{2\sqrt{5}}{3} $

d). \[-\dfrac{2\sqrt{5}}{3}\]

If chord PQ subtends an angle $ \theta $ at the vertex of $ {{y}^{2}}=4ax $ , then $ \tan \theta = $

a). $ \dfrac{2\sqrt{7}}{3} $

b). $ -\dfrac{2\sqrt{7}}{3} $

c). $ \dfrac{2\sqrt{5}}{3} $

d). \[-\dfrac{2\sqrt{5}}{3}\]

Two tangents on a parabola are $x-y=0$ and $x+y=0$. If $\left( 2,3 \right)$ is the focus of the parabola, then find the equation of the tangent at the vertex.

(a) $4x-6y+5=0$

(b) $4x-6y+3=0$

(c) $4x-6y+1=0$

(d) $4x-6y+\dfrac{3}{2}=0$

(a) $4x-6y+5=0$

(b) $4x-6y+3=0$

(c) $4x-6y+1=0$

(d) $4x-6y+\dfrac{3}{2}=0$

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