Question

a circle of radius 4 is concentric with the ellipse 3x*2+13y*2=78.prove that a common tangent is inclined to major axis at an angle is 45 degrees

Answer 1

Given: A circle with radius 4.

To prove: common tangent is inclined to major axis at 45 degree angle.

Proof: Ellipse equation is S = 3x^2+13y^2=78.

                                             = x^2/26 + y^2/6 = 1--------------(Standard form)

A circle with r = 4 is concentric to S.

So, S1= x^2 + y = 16.

equation of the tangent to S having slope 'm' is :

                                           y= mx root a^2m^2 + b^2----------(i)

                           1ar dist. from C(0,0) to P on (i) = radius = 4

                 d = |m(0) - 1(0) root a^2m^2+b^2|

                    ---------------------------------------------- = 4

                             root m^2+ (-1)

                       |0 - 0 root 26m^2 + 6

                   = -------------------------------- = 4

                            root m^2+1

On squaring both the sides we get

26m^2 + 6 = 16(m^2 + 1)

10m^2 = 10  ----------------------------------- m =+1 or -1

So, the value of slope of tangents is +1 and -1

m= tanß = tan45°

Hence, the inclination of tangent on major axis is 45°

Answer 2

Answer: The inclination of tangent on major axis is 45° proved.

Given:

• A circle with radius 4.

• Ellipse $3x^2+13y^2=78$.

To Find: Prove common tangent is inclined to major axis at 45 degree angle.

Step-by-step explanation:

Step 1: Ellipse equation is

$S = 3x^2+13y^2=78\\ = x^2/26 + y^2/6 = 1-(Standard \ form)$.

A circle with r = 4 is concentric to S.

So, $S_1= x^2 + y^2 = 16$.

Equation of the tangent to S having slope 'm' is :

$y= mx \sqrt {a^2m^2 + b^2}$----------(i)

Large distance from C(0,0) to P on equation (i) = radius = 4

$d = \frac{|m(0) - 1(0) \sqrt{a^2m^2+b^2}|}{\sqrt{m^2+(-1)^2} } =4\\\\d=\frac{|0 - 0 \sqrt{26m^2 + 6}|}{\sqrt {m^2+1}}$                            

Step 2: On squaring both the sides we get,

$26m^2 + 6 = 16(m^2 + 1)\\10m^2 = 10\\m =+1 or -1$

So, the value of slope of tangents is +1 and -1.

$m= tan\theta = tan45^o$

Hence, the inclination of tangent on major axis is 45°.

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