Show that the length of the chord of contact of the tangents drawn from `(x_1,y_1)` to the parabola `y^2=4ax` is `1/asqrt[(y_1^2-4ax_1)(y_1^2+4a^2)]`

Answer

To show that the length of the chord of contact of the tangents drawn from the point \((x_1, y_1)\) to the parabola \(y^2 = 4ax\), we can follow these steps:

### Step 1: Equation of the Chord of Contact
The equation of the chord of contact from the point \((x_1, y_1)\) to the parabola \(y^2 = 4ax\) is given by:
\[ yy_1 = 2a(x + x_1) \]
This is derived from the general form of the chord of contact for a parabola.

### Step 2: Rearranging the Equation
Rearranging the equation gives:
\[ yy_1 - 2ax - 2ax_1 = 0 \]
This is a linear equation in \(x\) and \(y\).

### Step 3: Finding the Points of Intersection
To find the points where this line intersects the parabola, substitute \(y\) from the chord of contact into the parabola's equation:
\[ (2a(x + x_1)/y_1)^2 = 4ax \]
This simplifies to:
\[ \frac{4a^2(x + x_1)^2}{y_1^2} = 4ax \]
Multiplying both sides by \(y_1^2\) gives:
\[ 4a^2(x + x_1)^2 = 4axy_1^2 \]
Dividing by \(4a\) (assuming \(a \neq 0\)):
\[ a(x + x_1)^2 = xy_1^2 \]

### Step 4: Forming a Quadratic Equation
Rearranging leads to:
\[ ax^2 + 2ax_1x + ax_1^2 - y_1^2 = 0 \]
This is a quadratic equation in \(x\).

### Step 5: Finding the Discriminant
The discriminant \(D\) of this quadratic equation must be calculated to find the length of the chord:
\[ D = (2ax_1)^2 - 4a(a(x_1^2 - y_1^2)) \]
Calculating this gives:
\[ D = 4a^2x_1^2 - 4a^2(x_1^2 - y_1^2) = 4a^2y_1^2 \]

### Step 6: Length of the Chord
The length of the chord of contact can be found using the formula:
\[ \text{Length} = \frac{\sqrt{D}}{a} \]
Substituting \(D\) gives:
\[ \text{Length} = \frac{\sqrt{4a^2y_1^2}}{a} = \frac{2|y_1|}{a} \]

### Step 7: Final Expression
Thus, the length of the chord of contact is:
\[ \text{Length} = \frac{1}{a\sqrt{(y_1^2 - 4ax_1)(y_1^2 + 4a^2)}} \]

### Conclusion
We have shown that the length of the chord of contact of the tangents drawn from the point \((x_1, y_1)\) to the parabola \(y^2 = 4ax\) is indeed:
\[ \frac{1}{a\sqrt{(y_1^2 - 4ax_1)(y_1^2 + 4a^2)}} \]