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Choose whether the given statement is true or false.

If two tangents to a parabola $y^{2} = 4 ax$ meet at an angle of $45 °$ , then the locus of their point of intersection is the curve $y^{2} - 4 ax = (x + a^{2})$ .

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True

False

answer is A.

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Detailed Solution

We are given the equation of parabola opened towards right as

y^{2} = 4 ax

......(1)
We know that the equation of any tangent to parabola (1) is given by

y = mx + \frac{a}{m}

.......(2)
Let (h, k) be the coordinates of the point of intersection of the tangents meeting at 45∘45∘ angle as given in the question. Since (h, k) lies on the tangent line (2) we have;

k = mh + \frac{a}{m}

⇒

h m^{2} - km + a = 0

The roots of the above equation say

m_{1}, m_{2}

will be the slopes of intersecting tangents. We use the sum of the roots formula and have;

m_{1} + m_{2} = \frac{- (- k)}{h} = \frac{k}{h}

We use product of the roots formula and have;

m_{1} m_{2} = \frac{a}{h}

We use the algebraic identity

{(a + b)}^{2} - {(a - b)}^{2} = 4 ab

for

a = m_{1}, b = m_{2}

to have;

{(m_{1 -} m_{2})}^{2} = {(m_{1} {+ m}_{2})}^{2} - 4 m_{1} m_{2}

⇒

{(m_{1 -} m_{2})}^{2} = {(\frac{k}{h})}^{2}

-4

(\frac{a}{h})

⇒

{(m_{1 -} m_{2})}^{2} = \frac{k^{2} - 4 a h}{h^{2}}

⇒

m_{1 -} m_{2} = \pm \frac{\sqrt{k^{2} - 4 a h}}{h}

We know that the angles between two lines θ with slopes

m_{1,} m_{2}

is obtained from the equation
tan

θ = \frac{m_{1 -} m_{2}}{1 + m_{1} m_{2}}

We put the given θ=45

°

an previously obtained

m_{1} m_{2 =} \frac{a}{h}

m_{1 -} m_{2} = \pm \frac{\sqrt{k^{2} - 4 ah}}{h}

to have;

\tan \tan 45 ° = \frac{\pm \frac{\sqrt{k^{2} - 4 a h}}{h}}{1 + \frac{a}{h}}

We square both side to have;
⇒

1 = \frac{k^{2} - 4 a h}{{(h + a)}^{2}}

We replace (h ,k) by ((x, y) to represent all such points of intersection of tangents

y = m_{1} x

\frac{a}{m_{1}}

y = m_{2} x

\frac{a}{m_{2}}

as the locus

y^{2} - 4 ax = {(x + a)}^{2}

Hence the given statement is proved.
So, the given statement is true.
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